Question

Write an inequality for the graph below.

Short answer

The inequality of the graph is $y>\frac{3}{4}x-1$.

Step-by-step solution

Step 1. State the concept of system of linear inequalities.

A system of linear inequalities in two variables consists of at least two linear inequalities in the same variables. The solution of a linear inequality is the ordered pair that is a solution to all inequalities in the system and the graph of the linear inequality is the graph of all solutions of the system.

The graph of an inequality in two variables is the set of points that represents all solutions to the inequality. A linear inequality divides the coordinate plane into two halves by a boundary line where one half represents the solutions of the inequality. The boundary line is dashed for > and < and solid for ≤ and ≥. The half-plane that is a solution to the inequality is usually shaded.

Step 2.  Find the equation of the dotted line.

In order to find the inequality first find the equation of the dotted line.

Equation of the dotted line can be found by two-point form.

Two–point form: Two point form is one of the important forms used to represent astraight line algebraically. The equation of a line represents each and every point on the line, i.e., it is satisfied by each point on the line. The two-point form of a line is used for finding theequation of a line given two points$\left(x_1,y_1\right)$ and$\left(x_2,y_2\right)$ on it.

The two-point form of a line passing through these two points is:

$y-y_1=\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}\times \left(x-x_1\right)\dots \left(1\right)$

From the graph see that the line passes through the points$\left(0,-1\right)$and $\left(4,2\right)$.

Substitute$\left(0,-1\right)$as$\left(x_1,y_1\right)$and$\left(4,2\right)$as $\left(x_2,y_2\right)$ in (1) to get the equation of the line.

$$\begin{gathered} y-y_1=\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}\times \left(x-x_1\right) \\ y-\left(-1\right)=\frac{\left(2-\left(-1\right)\right)}{\left(4-0\right)}\times \left(x-0\right) \\ y+1=\frac{\left(2+1\right)}{4}\times \left(x\right) \\ y+1=\frac{3}{4}\times \left(x\right) \\ y+1=\frac{3}{4}x \\ y=\frac{3}{4}x-1 \end{gathered}$$

is the equation of the dotted line.

Step 3. Write the inequality of the graph.

The shaded portion is above the line and the line is dotted one.

Therefore, inequality expression will be: $y>\frac{3}{4}x-1$.